A planar graph consisting of strings of variable densities is considered. The spectrum of the Dirichlet problem on the graph and the values of derivatives of the normalized eigenfunctions at the boundary vertices constitute the spectral data. The inverse problem is to recover the structure of the graph and the densities from the spectral data. If the graph doesn't contain cycles (is a tree), it is determined by the spectral data up to a natural isometry on the plane (Belishev, 2004). In the paper this uniqueness result is supplied with an efficient procedure of recovering the tree. The numerical illustration is presented.